Question

What is the value of \[\sin \left( \dfrac{7\pi }{4} \right)\]?

Answer 1

Hint: To solve the problem we have to find the value of \[\sin \left( \dfrac{7\pi }{4} \right)\]. For that we have to convert it as \[\left( 2\pi -\dfrac{\pi }{4} \right)\]. After that by using the double angle formula we can find the value of \[\sin \left( \dfrac{7\pi }{4} \right)\]. By using the double angle formula, we can easily find the value of \[\sin \left( \dfrac{7\pi }{4} \right)\].

Complete step-by-step solution:
For the given problem we have to find the value of \[\sin \left( \dfrac{7\pi }{4} \right)\].
Find the value of \[\sin \left( \dfrac{7\pi }{4} \right)\] using the double angle formula \[\sin \left( \dfrac{\pi }{4} \right)\]
Consider the given equation as equation (1)
\[a=\sin \left( \dfrac{7\pi }{4} \right)---(1)\]
By the trigonometry table of special arcs and until circle, we can write the equation (1) as
\[a=\sin \left( 2\pi -\dfrac{\pi }{4} \right)\]
By remembering the property of trigonometry is that \[\sin \left( 2\pi -\theta \right)=-\sin \left( \theta \right)\]we use this and apply it in the above equation.
\[a=\sin \left( -\dfrac{\pi }{4} \right)\]
As we know that \[\sin \left( \theta \right)=-\sin \left( \theta \right)\]substitute this property on above equation we get:
\[a=-\sin \left( \dfrac{\pi }{4} \right)---(2)\]
Finding \[\sin \left( \dfrac{\pi }{4} \right)\]by using the trigonometry identity
As we know the trigonometry identity
\[2{{\sin }^{2}}\left( a \right)=\left( 1-\cos \left( 2a \right) \right)\]
For finding the value of \[\sin \left( \dfrac{\pi }{4} \right)\]and also apply the formula we get:
\[2{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=\left( 1-\cos \left( 2.\dfrac{\pi }{4} \right) \right)\]
By simplifying it we get:
\[2{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=\left( 1-\cos \left( \dfrac{\pi }{2} \right) \right)\]
As we know that \[\cos \left( \dfrac{\pi }{2} \right)=0\]substitute this value on above equation we get:
\[2{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=\left( 1-0 \right)\]
By simplifying this we get:
\[2{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=1\]
Now we have to get the value from above equation that is
\[{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=\dfrac{1}{2}\]
By squaring on both sides, we get:
\[\sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}\]
Multiply \[(-)\]on both sides we get:
\[-\sin \left( \dfrac{\pi }{4} \right)=\dfrac{-1}{\sqrt{2}}\]
Now, the above value is substituted on equation (2) we get:
\[a=\dfrac{-1}{\sqrt{2}}\]
So, therefore value of \[\sin \left( \dfrac{7\pi }{4} \right)\]is \[\dfrac{-1}{\sqrt{2}}\].

Note: All trigonometry tables and trigonometric formulas must be understood by students. While squaring the equation \[{{\sin }^{2}}\left( \dfrac{\pi }{4} \right)=\dfrac{1}{2}\]. We will receive two cases, one affirmative and one negative, but the negative answer will be dismissed. Because \[\sin \left( \dfrac{\pi }{4} \right)\]is positive. This problem can be solved using the sine half angle formula too.